应用泛函分析

Preface

Contents of AMS Volume 108

1　The Hahn-Banach Theorem Optimization Problems

1.1 The Hahn-Banach Theorem

1.2 Applications to the Separation of Convex Sets

1.3 The Dual Space C[a, b]*

1.4 Applications to the Moment Problem

1.5 Minimum Norm Problems and Duality Theory

1.6 Applications to Cebysev Approximation

1.7 Applications to the Optimal Control of Rockets

2 Variational Principles and Weak Convergence

2.1 The nth Variation

2.2 Necessary and Sufficient Conditions for Local Extrema and the Classical Calculus of Variations

2.3 The Lack of Compactness in Infinite-Dimensional Banach Spaces

2.4 Weak Convergence

2.5 The Generalized Weierstrass Existence Theorem

2.6 Applications to the Calculus of Variations

2.7 Applications to Nonlinear Eigenvalue Problems

2.8 Reflexive Banach Spaces

2.9 Applications to Convex Minimum Problems and Variational Inequalities

2.10 Applications to Obstacle Problems in Elasticity

2.11 Saddle Points

2.12 Applications to Dui~lity Theory

2.13 The von Neumann Minimax Theorem on the Existence of Saddle Points

2.14 Applications to Game Theory

2.15 The Ekeland Principle about Quasi-Minimal Points

2.16 Applications to a General Minimum Principle via the Palais-Smale Condition

2.17 Applications to the Mountain Pass Theorem

2.18 The Galerkin Menhod and Nonlinear Monotone Operators

2.19 Symmetries and Conservation Laws (The Noether Theorem

2.20 The Basic Ideas of Gauge Field Theory

2.21 Representations of Lie Algebras

2.22 Applications to Elementary Particles

3 Principles of Linear Functional Analysis

3.1 The Baire Theorem

3.2 Application to the Existence of Nondifferentiable Continuous Functions

3.3 The Uniform Boundedness Theorem

3.4 Applications to Cubature Formulas

3.5 The Open Mapping Theorem

3.6 Product Spaces

3.7 The Closed Graph Theorem

3.8 Applications to Factor Spaces

3.9 Applications to Direct Sums and Projections

3.10 Dual Operators

3.11 The Exactness of the Duality Functor

3.12 Applications to the Closed Range Theorem and to Fredholm Alternatives

4 The Implicit Function Theorem

4.1 m-Linear Bounded Operators

4.2 The Differential of Operators and the Fr~chet Derivative

4.3 Applications to Analytic Operators

4.4 Integration

4.5 Applications to the Taylor Theorem

4.6 Iterated Derivatives

4.7 The Chain Rule

4.8 The Implicit Function Theorem

4.9 Applications to Differential Equations

4.10 Diffeomorphisms and the Local Inverse Mapping Theorem

4.11 Equivalent Maps and the Linearization Principle

4.12 The Local Normal Form for Nonlinear Double Splitting Maps

4.13 The Surjective Implicit Function Theorem

4.14 Applications to the Lagrange Multiplier Rule

5 Fredholm Operators

5.1 Duality for Linear Compact Operators

5.2 The Riesz-Schauder Theory on Hilbert Spaces

5.3 Applications to Integral Equations

5.4 Linear Fredholm Operators

5.5 The Riesz-Schauder Theory on Banach Spaces

5.6 Applications to the Spectrum of Linear Compact Operators

5.7 The Parametrix

5.8 Applications to the Perturbation of Fredholm Operators

5.9 Applications to the Product Index Theorem

5.10 Fredholm Alternatives via Dual Pairs

5.11 Applications to Integral Equations and Boundary-Value Problems

5.12 Bifurcation Theory

5.13 Applications to Nonlinear Integral Equations

5.14 Applications to Nonlinear Boundary-Value Problems

5.15 Nonlinear Fredholm Operators

5.16 Interpolation Inequalities

5.17 Applications to the Navier-Stokes Equations References

List of Symbols

List of Theorems

List of Most Important Definitions

Subject Index

More precisely， by （i）， I mean a systematic presentation of the materialgoverned by the desire for mathematical perfection and completeness ofthe results. In contrast to （i）， approach （ii） starts out from the question"What are the most important applications？" and then tries to answer thisquestion as quickly as possible. Here， one walks directly on the main roadand does not wander into all the nice and interesting side roads.
　　The present book is based on the second approach. It is addressed toundergraduate and beginning graduate students of mathematics， physics，and engineering who want to learn how functional analysis elegantly solvesma~hematical problems that are related to our real world azld that haveplayed an important role in the history of mathematics. The reader shouldsense that the theory is being developed， not simply for its own sake， butfor the effective solution of concrete problems.